Solving Problems on Special Classes of Graphs

Abstract

Abstract Most papers which solve a problem on a graph class assume there is a promise that the input is in the class. Thus, the algorithm could produce incorrect output, or fail to terminate in the time bound claimed, if the input is not in the class. We propose a stricter notion of solving problems on special classes of graphs. If a graph G is not in the class, the algorithm may either answer the problem correctly, or announce that G is not in the class. We call such an algorithm a robust algorithm. This talk describes robust algorithms for problems on classes where no robust algorithm was known previously, shows that some problems which can be solved on classes in polynomial time given a promise that the graph is in the class are intractable for robust algorithms, and poses open problems. We show that the clique problem on unit disk graphs can be solved robustly. Previous solutions needed a geometric model as input. We also show that no robust polynomial algorithm solves independent set on well-covered graphs unless P = NPM.